Spring Cylinder

One part of a cylinder is filled with one mole of a monatomic ideal gas at a pressure of 1 atm and temperature of 300 K. A massless piston separates the gas from the other section of the cylinder which is evacuated but has a spring at equilibrium extension attached to it and to the opposite wall of the cylinder. The cylinder is thermally insulated from the rest of the world, and the piston is fixed to the cylinder initially and then released (see Figure 1.1). After reaching equilibrium, the volume occupied by the gas is double the original. Neglecting the thermal capacities of the cylinder, piston, and spring, find the temperature and pressure of the gas.

Figure 1.1

SOLUTION

For a thermally insulated system (no heat transfer), the energy change ∆ε is given by

(1)

Figure 1.2

where 0 and 1 correspond to the initial and final equilibrium states of the system, with sets of parameters T₀, P₀, V₀ and T₁, P₁, V₁ respectively. In this case, the gas is expanding, therefore some positive work is done by the gas, which indicates that the energy change is negative, and the temperature decreases. ∆ε for an ideal gas depends only on the change in temperature:

(2)

Where C_v is the heat capacity of one mole of the gas at constant volume (for a monatomic gas C_v = 3R/2). The work done by the gas goes into compressing the spring:

(3)

where K is the spring constant and x is the change of the piston position (see Figure 1.2). On the other hand, when equilibrium is reached, the compression force of the spring

(4)

where A is the cross section of the piston. So

(5)

where we used the ideal gas law for one mole of gas. Substituting (5) into (3), we have

(6)

Notice that Ax is the volume change of the gas:

(7)

Therefore,

(8)

Substituting (8) and (2) into (3), we obtain

(9)

and

(10)

The temperature indeed has decreased. As for the pressure, we have for the initial state

(11)

Now V₁ = 2V₀, so

(12)

and